Conductance measurement of a conical tube and calculation of the pressure distribution

Conductance measurement of a conical tube and calculation

of the pressure distribution

B. Merciera兲

Laboratoire de l’Accélérateur Linéaire (LAL), Université Paris-Sud, Batiment 200, BP34, 91898 Orsay Cedex, France

共兲

In recent articles关J. Gomez-Goni, and P. J. Lobo J. Vac. Sci. Technol. A 21, 1452 共2003兲; P. Swemin and M. Niewinski, Vacuum 67, 359共2002兲兴 the conductance of conical tube in the molecular flow regime has been calculated using the Monte Carlo method or by the resolution of the Clausing integral equation, reformulated by Iczkowski et al.关J. Phys. Chem. 67, 229 共1963兲兴, for the case of a cone. The comparison between the analytical values and different simulations allows one to determine a correction factor k to apply to the intrinsic conductance of the cones. This coefficient depends on the propagation direction of flow and increases considerably for larger conic angles. For a cone half-angle of 40° and a length ten times greater than smallest entrance radii, the correction factor is approximately 5.3 for a circulating flow from the smallest to the largest orifice. Our experimental device measured the conical conductance by a dynamic method. In order to do this, it was necessary to determine the surface pressure distribution. The extension of the Oatley method, with the addition of several components of various transmission probabilities, permits one to establish this distribution for a vacuum system and thus to give the pressure measured by a gauge situated along the wall of the duct. This method provides a good approximation for tubes and cones and can be used for engineering practice. The determination of this distribution is all the more critical when the conductance and the pumping speed are large and can thus have a great influence on the vacuum metrology.

关兴

I. INTRODUCTION

The first analytical calculations of conductance were car-ried out by Knudsen using the following expression:

CL= 1 4VmSW with W = 16 3 S UL, 共1兲

where Vm represents the average velocity of the molecules for a Maxwell-Boltzmann distribution, S the cross sectional area of the inlet aperture, U the perimeter, and L the length of the duct. However, this formula is valid only for infinitely long tubes compared to their diameters. Smoluchowski1 showed that this equation is incorrect and that it is necessary to replace it by CL= 1 4Vm A 2L with A =

冕

S

冕

−␲/2 +␲/2₁ 2␳ 2_{cos共␪兲d␪ds, 共2兲}

where␳is a cord of the section making an angle␪ with the normal to the perimeter s. The resolution of the integral in-troduces a correction factor k into Knudsen’s formula,

W =16 3 S U k L.

This coefficient k takes into account the different geometry of the inlet aperture. This expression remains valid for an infinitely long tube conductance. To take into account a short

tube conductance, Dushman combines CL with an entrance conductance Ce共for L→0 then C→Ce兲.

C = CLCe CL+ Ce

. 共3兲

In the case of an infinitely large upstream chamber, Ce = 1 / 4VmS. Many references

1,2

give the expression for the conductance value for sections which vary with their lengths, by integrating Knudsen’s equation.

CL= 4 3Vm 1 兰0 L_共U/kS2_兲dl. 共4兲

However, the equations above are only approximations and it is often necessary, when the geometrical forms become com-plex, to determine the conductance by the Monte Carlo method or by solving the numerical integral of Clausing.3 These two methods allow one to calculate the transmission probability of the molecules which corresponds to the factor

W. For a cylindrical tube, the relative error between the

Dushman equation and the Clausing equation reaches 12% and it can become higher than 50% for complex geometries. II. CALCULATION OF THE CONICAL

CONDUCTANCE A. Analytical calculation

We consider a conical tube共Fig. 1兲 of length L, R0, and Rk being the smallest and largest radii, respectively, and ␣ its half-angle. We distinguish two intrinsic conductances

ac-a兲_{Author to whom correspondence should be addressed; electronic mail:}

mercier@lal.in2p3.fr

cording to the flow direction. “Direct” conductance CA for flow circulating from R0towards Rkand “back” conductance

CR for flow in the opposite direction.

The integration of Eq.共4兲, with k supposed to be constant for a given angle, is expressed by

CA= 4 3␲Vm R₀2RK 2 k 共R0+ RK兲L . 共5兲

The conductance value CRcan be calculated by applying the Dushman equation and the equilibrium condition of flows 1 / CR+ 1 / Ce,R= 1 / CA+ 1 / Ce,A, we obtain

␥=CA

CR

= 1 +16

3 k tan共␣兲. 共6兲

We will determine this correction factor k by the Monte Carlo method.

B. Monte Carlo method

Gomez-Goni and Lobo4and Swemin and Niewinski5 car-ried out a comparison between the Monte Carlo method and the numerical resolution of the Clausing integrals. These in-tegrals were reformulated by Iczkowski6 for the case of the conical tube. This study made it possible to validate the method of resolution by obtaining very small relative varia-tions. The transmission probability by Monte Carlo method is calculated with the following assumptions: 共1兲 the flow must be molecular, 共2兲 the adsorption of molecules on the chamber is neglected, 共3兲 the law of emission of the mol-ecules is assumed to follow a cosine distribution共Lambert’s law兲, and 共4兲 collisions with the walls are subject to diffuse reflection 共cosine law兲. Assumption 共3兲 supposes that the upstream chamber is infinitely large and that its speed distri-bution is of the Maxwell-Boltzmann type. Under these conditions,

CM=

1

4VmSWM= CeWM. 共7兲

Wmrepresents the transmission probability of the molecules calculated by the Monte Carlo method. According to Eqs.共5兲 and共7兲, and by taking account of the entrance conductance, we can express the correction factor k as a function of W. For the case of the direct cone, we get

1 CM,A = 1 Ce,A + 1 CA , k = 3 16 WM,A 1 − WM,A 1 R0/L + tan共␣兲

冋

1 1 +共L/R0兲tan 共␣兲 + 1

册

. 共8兲 The values of W_M,Awere determined by various programs: a Monte Carlo 共MC兲 program coming from CERN,7 a direct simulation Monte Carlo共DSMC兲 program of Bird,8and the data published in the article of Gomez-Goni and Lobo.4 Ac-cording to Fig. 2, the correction factor k becomes consider-able for large angles and translates the fact that the reflection of the particles is diffuse compared to the normal of the wall surface and thus is a function of the cone slope. For example, a direct cone of ratio L / R0= 10 and angle ␣= 40° has a

cor-rection factor of approximately 5.3 and consequently the in-trinsic backconductance of the cone will be approximately 25 times lower than that for the direct direction.

III. EXPERIMENTAL MEASUREMENTS A. Experimental device

The measurement of the conductance is performed using a dynamic method. It consists of injecting a known flow Q, by the intermediary of a fine capillary of conductance C1, and

measuring pressure differential on both sides of the conduc-tance CR,1= Q /共PJAB1− PJAB2兲.

Figure 3 represents the experimental device, with an input flow line consisting of a chamber of volume V. This volume is maintained with a work pressure P0by an adjustable leak

and this pressure is measured using capacitance diaphragm gauges共Baratron 627A MKS of 100 and 1 Torr full scale兲. This volume is connected to a fine capillary with a diameter of 0.8 mm and a length of 510 mm. The dimensions of the cones CR,1 and CA,2 are expressed in millimeter and their half-angles are 40.28° and 38.55°, respectively. Special

at-FIG. 1. Conical tube scheme.

tention was paid so that the half-angle is constant over the entire length of the cones and so that the angles on the level of the junction of the two cones and the junction with the

JAB2 gauge tube are sharp. The diameter of the junction of the two cones is 20 mm± 0.1 mm. The measurement of the pressure is carried out using two identical spinning rotor gauges共Viscovac of Leybold兲. The second pumping installa-tion is composed of a hybrid turbomolecular pump 共TV141 Navigator of Varian兲, nominal pumping speed for nitrogen

S2= 125 l s−1, and a diaphragm pump Vacuubrand MZ2D of

1.9 m3_h−1_{. Later, a prepumping elbow, CC, was added. The}

intrinsic conductance was 359 l s−1 _{共unless specified, the}

values of conductance or flow are given for nitrogen at 20 ° C兲.

B. The measurement of flow

The measurement of flow involves the use of the small capillary conductance; it is thus important to know its value. We used the method of the pressure evolution for the volume

V. This method consists in measuring the time ⌬t to pass

from a pressure P₀ to P_0,t. The following assumptions are necessary; outgassing from the walls of the introduction chamber and from the capillary is negligible as compared to the introduced flow and it is considered that the temperature is constant. With the ideal gas law, we get

VdP0

dt = − P0C1. 共9兲

The conductance C₁ depends on the flow regime. The full curve, the circles, and crosses of Fig. 4共b兲 represent, respec-tively, the theoretical value given by Knudsen and the value measured by C1=共−V/ P兲共⌬P兲/共⌬t兲 by assuming a weak

variation of P.

There is a good agreement between the measurement and the theory. The divergence noted while approaching the mo-lecular mode can be explained by the modification of the equivalent length due to the capillary’s elbow. In our case, to make precision measurements, it is necessary that the pres-sures measured by the spinning rotor gauges are higher than 5⫻10−6 _{mbar. For that, in comparison with the conductance}

value of the prepumping elbow, CC, nominal pumping speed

S2 and conductance CR,1= 35.3 l s−1and CA,2= 798 l s−1, the

pressure P0 in the volume of introduction will have to be

higher than 1 mbar. CR,1and CA,2are calculated by using the Knudsen equation and with a correction factor k, respec-tively, of 5.3 and 4.5共see Fig. 2兲. Before the nitrogen intro-duction, the system pressure is about 5⫻10−7Pa. Table I recapitulates the measurements taken with and without the presence of the prepumping elbow. The principal measure-ment error comes from the difficulty in maintaining P0

con-stant, and consequently the accuracy measurements of pres-sure for the spinning rotor gauges are estimated to within ±5%.

The first two lines of the Table I are values with the pres-ence of the prepumping elbow, CC, and the remaining lines with the direct connection of the turbo pump. The measure-ment of these values was taken at a temperature of 25 ° C. The difference in the value of the intrinsic conductance of cone C_R,1 with and without the prepumping elbow is rela-tively small. The modification of the emission law of the molecules from the turbo-pump seems to influence only moderately the value of CR,1. The variation between the cal-culated conductance共35.3 l s−1_{兲 and measured one 共69 l s}−1_兲 FIG. 3. Experimental device.

FIG. 4. The conductance C1 as a function of the upstream pressure.

TABLEI. “Back” cone measurement.

is explained by the fact that first one is calculated compared to the pressures upstream and downstream 共entry and exit surfaces兲 while the second one is measured compared to the tube surface. Moreover, in our case, it is necessary to take into account the position of the gaugeJAB1, between the two cones.

IV. CALCULATION OF THE SURFACE DISTRIBUTION OF PRESSURE

A. The case of the cylindrical tube

Initially, to calculate the collision number on a surface element, ds, one uses the Clausing PCl probability of the

number of molecules crossing without collision.

Pcl=

冕

0 ␪max

2 cos共␪兲sin共␪兲S共␪兲d␪, 共10兲

with S共␪兲 the transmission probability of the molecules with-out collision for a given angle␪ 关Fig. 5共a兲兴.

S共␪兲 =2

␲关arccos共y兲 − y共1 − y2兲1/2兴

with y =关L/共2r兲兴tan共␪兲, 共11兲

thus for a length L→dx and a cosine distribution law of the molecules, the number of collisions for N emitted molecules is

Nchoc= N共1 − Pcl兲 = N

冉

dx

r

冊

. 共12兲

To determine the surface pressure at the beginning of a cir-cular conductance, it is possible to use the Oatley method, by associating a tube of length dx and of transmission probabil-ity W1= Nchoc/ 2, to a second length L of probability W2关see

Fig. 5共b兲兴.

At each passage, whatever the direction, one assumes that the number of hits on the surface 2␲rdx is Ndx / r. It is

ob-vious that this is only an approximation, not taking into ac-count the “beam” effect, with variations of molecular spatial distributions for different crossings. In addition, in order to

be able to simulate a flow, the rejected molecules are injected into the circuit, at the entrance of W₁. By assuming that W₁is close to 1, we obtain

N_choc= Ndx

r

冉

2 − W₂

W₂

冊

. 共13兲

Consequently, the surface pressure for dx→0 can be ex-pressed by PS₀= Q 共1/4兲Vm␲r2

冉

2 − W2 2W2

冊

. 共14兲

By deriving the Q =␦CdP equation 共with ␦C the intrinsic

conductance for a length dx兲 and while taking as initial con-dition Ps₀共obtained by the Oatley method兲, the surface pres-sure distribution, for a steady flow Q and for a perfect pump 共sticking coefficient=1兲, we have

PS共x兲 = − Q CLL x + PS₀ = − Q 1 4Vm␲r 2_W 2

冋

1 2+共1 − W2兲

冉

1 2− x L

冊

册

. 共15兲

Helmer and Levi9obtained the same result by using a differ-ential form of the Oatley equation and by taking 1 / W₂= 1 + 1 / f with f = 2r / L as form factor. Figure 6 compares the pressure distribution calculated by Eq.共15兲, by the Helmer-Levi method, and also by the Monte Carlo method 共MC or DSMC兲, for tubes of radius 0.02 m and length-radius ratios of 3 and 30. The flow Q is 2.8⫻10−5 _{Pa m}3_s−1_{for nitrogen}

at 20 ° C. The transmission probabilities W2are, respectively,

0.421 and 0.0797. For the short tube, we observe a good correlation between the P0关=Q/共CeW2兲兴 of the upstream

chamber and surface pressure Ps₀. According to Eq.共15兲, this difference is constant and independent of W2,⌬P= P0− Ps₀ = Q /共2Ce兲 and it also applies to the difference in pressure downstream PSL− PL 共in our case PL= 0兲. This variation of pressure between volume and surface seems to be a good

FIG. 5. Cylindrical tube scheme 共a兲 on a surface element ds; 共b兲 on all

surface tube. F_{at 20 ° C, with a flux of 2.8}IG. 6. Surface pressure distribution for cylindrical conductance for nitrogen_⫻10−5_{Pa m}3_s−1_{, for total absorption共a兲 for}

approximation in spite of the fact that the change of the molecular spatial distribution is not taken into account. For the long tube, the⌬P becomes negligible with respect to P0

共⌬P/ P0= W2/ 2 with small W2兲, and the difference with the

Helmer-Levi method arises from the fact that, according to a number of authors, the form factor f for a long tube tends towards 8r /共3L兲.

It is possible to extend the method of Oatley in order to take into account a pumping capacity S0= Ce␥, with Ce the entrance conductance corresponding to the inlet pump and␥ the sticking coefficient on this surface. The collision number on the element dx becomes

N_choc= Ndx r

冋

2 − W₂ W2 +2共1 −␥兲 ␥

册

. 共16兲

With the same reasoning as previously, the surface distribu-tion of pressure is expressed by

PS共x兲 = Q CeW2

冋

1 2+共1 − W2兲

冉

1 2 − x L

冊

+ W₂共1 −␥兲 ␥

册

. 共17兲 The upstream and downstream pressures are expressed clas-sically by, respectively, P0= Q /共CeW2兲关1+W2共1−␥兲/␥兴 and PL=共Q/Ce兲关共1−␥兲/␥兴 with P0− PL= Q /共CeW2兲. Figure 7

represents the pressure distribution on the tube surface for the L / r ratio and for different sticking coefficients. Equation 共17兲 gives, to a good approximation, the pressure measured by a gauge for a system of pumping connected by a tube to a large chamber. It should be noted that the difference be-tween the surface pressures and the upstream and down-stream pressures is constant as previously whatever the tube length and the pumping speed. It is equal to ⌬P= P0− Ps₀ = PSL− PL= Q /共2Ce兲.

B. The case of a conical tube

In the same way, to calculate the collision number on an surface element ds, one uses the probability of Clausing,

P_cone, of the number of molecules crossing without collision.

Pcone=

冕

0

␪max

2 cos共␪兲sin共␪兲Sc共␪兲d␪

with␪max= arctan关共Rk− R0兲/L兴, 共18兲

with Sc共␪兲 the transmission probability of the molecules without collision for a given angle␪and the direct direction of the cone.

Sc共␪兲 = 1

␲

再

arccos共y兲 − y共1 − y2兲1/2 +RK 2 R₀2关arccos共y1兲 − y1共1 − y1 2_兲1/2_兴

冎

_, with y =关L/共2R0兲兴tan共␪兲+共R0 2 − Rk 2_兲/关2LR 0tan共␪兲兴 and y1 =关L/共2Rk兲兴tan共␪兲+共Rk 2

− R₀2兲/关2LRktan共␪兲兴. For ␪ ranging between 0 and␣, Sc共␪兲=1. This equation can be solved nu-merically. For a length L→dx, a cosine distribution law for the collision number for N emitted molecules is Nchoc= N共1

− Pcone兲. According to Fig. 8, we can express this collision

number as a function of the cone half-angle,␣.

Nchoc= N2 dx R0 tan共␣兲f共␣兲 with f共␣兲 = 1 2 tan共␣兲

冉

1 − 8 3 ␣ ␲+ 4 3 ␣2 ␲

冊

. 共19兲

With the same reasoning, the collision number for a cone in the back direction is

Nchoc−r= N2 dx

R₀tan共␣兲关f共␣兲 + 1兴. 共20兲

To determine the surface pressure, in the same way as that for a tube, it is possible to extend the Oatley method again,

FIG. 7. Surface pressure distribution for cylindrical conductance for nitrogen at 20 ° C, with a flux of 2.8⫻10−5_{Pa m}3_s−1_{,共a兲 for different sticking}

coef-ficients␥= 0.8 and 0.5 and for L / r = 3 with r = 2 cm共W2= 0.421兲; 共b兲 for a

sticking coefficient␥= 0.8 and for L / r = 1 with r = 2 cm共W2= 0.672兲.

FIG. 8. Collision number on the cone surface for L→dx 共10−5_{m兲 and for}

by associating several cones of transmission probability in the direct W and back Q directions 关see Fig. 9兴. For each passage of molecules, one enters the collision number on the surface of the length element dx while applying, according to the direction, Eq. 共19兲 or 共20兲. The probabilities W2 and Q2

tend towards 1 and by simulating a flow and a capacity of pumping, we get a total collision number according to x, for an element dx,

共21兲

with Q3/ W3= R2共x兲/Rk2, W3is of course a function of x and

can be expressed on the basis of Eqs.共6兲 and 共7兲 by 1 W3 = 1 + 关R共x兲 + Rk兴 16 3 Rk 2 k关L − x/R共x兲兴 共L − x兲 and R共x兲 = R₀+ x tan共␣兲,

the factor k varies slightly according to L / R0共see Fig. 2兲, for

an angle of 40°, k varies from 4.5 to 5.3 for, respectively, a

L / R0= 1 and L / R0= 10. We can roughly express this

varia-tion as an equavaria-tion. For a direct cone of 40°, k共t=L/R0兲

⬇共4.5+2.65t+2.72t2兲/共1+0.56t+0.5t2兲.

1. Calculation of the surface distribution by the method known as the extension of Oatley

By estimating that the number of shocks per unit time on the surface of the cone length dx is⌫=共1/4兲VmnS共dx兲 共with

n molecular density兲, the surface distribution of pressure is

expressed as PS₁共x兲 ⬇ QN_choc 共1/4兲VmS共dx兲N = Q sin共␣兲 共1/4兲Vm␲R共x兲2 h. 共22兲

2. Calculation of the surface distribution by the traditional method_{„Knudsen…}

As in the case of the tube, by deriving the Q =␦CdP

equa-tion 共with␦C intrinsic cone conductance given by Knudsen

and for a length dx兲 and while taking Ps₁共0兲 as initial condi-tion 共obtained by the Oatley method兲, this distribution of surface pressure can be written as follows:

PS₂共x兲 =

冕

− Q 共2/3兲Vm␲k dx R共x兲3 = Q 共4/3兲Vm␲k tan共␣兲

冋

1 R共x兲2− 1 R₀2

册

+ PS1共0兲. 共23兲

3. Extension of the Oatley method for a cone in the back direction

In the same manner, we can determine the surface pres-sure distribution for a cone in the back direction, with, in this case, R共x兲=Rk− x tan共␣兲, a corrective factor k

⬘

= k /关1 +共16/3兲k tan共␣兲兴 and with a collision number,

Nchoc−R= N 2dx tan共␣兲 R共x兲

再

f共␣兲

冉

1 − Q3 Q3

冊

+

冋

1 + f共␣兲 Q3

册

+W3 Q3 共1 −␥兲共2 − f共␣兲 + 1兲 ␥

冎

. 共24兲

It is obvious that this is only an approximation which does not take into account the beam effects 共the changes of the spatial distribution of the molecules for the different pas-sages兲. According to Fig. 10, one can note that Eq. 共23兲 derived from the conductance calculated by Knudsen is not valid, or only for one correction factor k equal to 1.

In this case, as in the extension of the Oatley method, we obtain a relatively good approximation of the surface pres-sure compared to simulation by Monte Carlo. This approxi-mation improves when the pumping speed decreases. Figure 10 shows the influence of a pump on the surface pressure distribution at the cone exit. This pump will modify the

spatial distribution of the molecules at the cone exit. In the case of the large sticking coefficient, the spatial distribution of the molecules at the cone exit is not Lambert’s law. For the back cone, the approximation by the extension of the Oatley method is satisfactory and that for all pumping speeds 共Fig. 11兲. The good agreement between these two methods encourages to consider the spatial distribution on the exit tube as Lambert’s law. The Knudsen equation is not very sensitive to the correction factor k and greatly diverges when approaching downstream opening and for high pumping capacities.

V. COMPARISON BETWEEN MEASUREMENT AND SIMULATIONS

The experimental setup共see Fig. 3兲 is formed by the junc-tion of a back cone and a direct cone. It is necessary for the extension of the Oatley method to determine the sticking coefficients corresponding to a turbo pumping speed of

S0= 125 l s−1, for the second cone␥= S0/ Ce2= 0.135 and for the first cone ␥1 = 0.768 first 关with CA,2= 798 l s−1 for k = 4.8 and 1 /共Ce1␥1兲=1/125−1/Ce2+ 1 / CA,2+ 1 / Ce1兴. By the Knudsen method, it is enough to fix the pressure at the entry of the first cone or the exit of the second with

FIG. 12. Surface pressure distribution on the two conical tubes of experi-mental device for a pumping speed of 125 l / s.

FIG. 10. Surface pressure distribution for conical conductance of “direct” direction for nitrogen at 20 ° C, with a flux 2.8⫻10−5_{Pa m}3_s−1_{, for different sticking}

coefficients, for L / R0= 10 with r = 1 cm and␣= 40.28°.

FIG. 11. Surface pressure distribution for conical conductance of “back” direction for nitrogen at 20 ° C, with a flux 2.8⫻10−5_{Pa m}3_s−1_{, for different}

P1⬇Q/共1/S0− 1 / Ce2+ 1 / CA,2+ 1 / CR,1兲 or P0= Q /共1/S0

− 1 / C_e2兲. For simulations by Monte Carlo, we simulate the two cones and as well as the turbo pump at the system exit as a unique system and we take as sticking coefficient of ␥= 0.135.

Figure 12 shows a good correspondence between simula-tion by Monte Carlo and Oatley. A small variasimula-tion is noted with the points of measurements taken without a prepumping elbow but remains within the error margin. The pressure re-corded by the two spinning rotor gauges seems to validate in our case the simulations methods and consequently to con-firm the correction factors k.

However, the formula for the surface pressure distribution derived from the Knudsen conductance with the adequate

k factors diverges to give a pressure at x = 0.1 m of 5.5

⫻10−3_{Pa instead of a measured pressure of 1.4⫻10}−2_Pa

and a simulated pressure of 1.55⫻10−2 _Pa.

VI. CONCLUSION

The Monte Carlo method made it possible to determine the correction factor k to apply to the Knudsen conductance formula in the case of the cone. In addition, the extension of the Oatley method gives a good approximation of the surface

pressure distribution for tubes or cones taking into account a pump at the exit of the conductance. This distribution is all the more sensitive since the lengths are short and the pump-ing capacities are important. This can, therefore, have a great influence on the vacuum metrology. Our measurements seem to support our simulations but they need to be improved and extended to other geometries.

ACKNOWLEDGMENT

The author would like to thank Dr. T. Garvey for improv-ing the English of the manuscript.

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